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Least squares solution in the presence of known covariance
x=lscov(A,b,V) [x,dx] = lscov(A,b,V)
x = lscov(A,b,V)
returns the vector x that solves A*x = b + e where e is normally distributed with zero mean and covariance V. Matrix A must be m-by-n where m > n. This is the over-determined least squares problem with covariance V. The solution is found without inverting V.
[x,dx] = lscov(A,b,V)
returns the standard errors of x in dx. The standard statistical formula for the standard error of the coefficients is:
mse = B'*(inv(V)-inv(V)*A*inv(A'*inv(V)*A)*A'*inv(V))*B./(m-n) dx = sqrt(diag(inv(A'*inv(V)*A)*mse))The vector
x minimizes the quantity (A*x-b)'*inv(V)*(A*x-b). The classical linear algebra solution to this problem is
x = inv(A'*inv(V)*A)*A'*inv(V)*b
lscov function instead computes the QR decomposition of A and then modifies Q by V.
\ Matrix left division (backslash)
nnls Nonnegative least squares
qr Orthogonal-triangular decomposition